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Journal of the London Mathematical Society 2000 62(1):149-160; doi:10.1112/S0024610700001034
© 2000 by London Mathematical Society
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© The London Mathematical Society

Definiteness of the Peano Kernel Associated with the Polyharmonic Mean Value Property

Werner Haußmann and Ognyan Kounchev

Department of Mathematics, Gerhard-Mercator-University Duisburg, 47048 Duisburg, Germany
Institute of Mathematics, Bulgarian Academy of Sciences 8 Acad. G Bontchev Str., 1113 Sofia, Bulgaria

Received 10 July 1998.

The definiteness of the Peano kernel is proved for a functional associated with the mean-value property of Picone and Bramble and Payne for polyharmonic functions in the ball. An important corollary of this is that if a function f satisfying (–1)p{Delta}pf>0 vanishes on p concentric spheres centered at 0, then f(0)>0. This generalizes a well-known property of subharmonic functions (which arise in the special case p = 1).


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